Further one immediately verifies that in this case the phase boundary is quite analogue to a contact discontinuity.
Therefore we use in our numerical simulations, for the case without phase transition, the wave speed estimates for the HLLC solver as given in Refs. [1]}, [2]}.
Given the (left) vapor state and the (right) liquid state we proceed as follows
\(S_L &= u_L + a_L,\\S_V & = u_V - a_V,\\w &= \frac{p_L - p_V + \rho _Vu_V(S_V - u_V) - \rho _Lu_L(S_L - u_L)}{\rho _V(S_V - u_V) - \rho _L(S_L - u_L)},\\\rho _L^\ast &= \rho _L\frac{S_L - u_L}{S_L - w}.\) 
